from sympy.assumptions.refine import refine from sympy.calculus.accumulationbounds import AccumBounds from sympy.concrete.products import Product from sympy.concrete.summations import Sum from sympy.core.function import expand_log from sympy.core.numbers import (E, Float, I, Rational, nan, oo, pi, zoo) from sympy.core.power import Pow from sympy.core.singleton import S from sympy.core.symbol import (Symbol, symbols) from sympy.functions.elementary.complexes import (adjoint, conjugate, re, sign, transpose) from sympy.functions.elementary.exponential import (LambertW, exp, exp_polar, log) from sympy.functions.elementary.hyperbolic import (cosh, sinh, tanh) from sympy.functions.elementary.miscellaneous import sqrt from sympy.functions.elementary.trigonometric import (cos, sin, tan) from sympy.matrices.expressions.matexpr import MatrixSymbol from sympy.polys.polytools import gcd from sympy.series.order import O from sympy.simplify.simplify import simplify from sympy.core.parameters import global_parameters from sympy.functions.elementary.exponential import match_real_imag from sympy.abc import x, y, z from sympy.core.expr import unchanged from sympy.core.function import ArgumentIndexError from sympy.testing.pytest import raises, XFAIL, _both_exp_pow @_both_exp_pow def test_exp_values(): if global_parameters.exp_is_pow: assert type(exp(x)) is Pow else: assert type(exp(x)) is exp k = Symbol('k', integer=True) assert exp(nan) is nan assert exp(oo) is oo assert exp(-oo) == 0 assert exp(0) == 1 assert exp(1) == E assert exp(-1 + x).as_base_exp() == (S.Exp1, x - 1) assert exp(1 + x).as_base_exp() == (S.Exp1, x + 1) assert exp(pi*I/2) == I assert exp(pi*I) == -1 assert exp(pi*I*Rational(3, 2)) == -I assert exp(2*pi*I) == 1 assert refine(exp(pi*I*2*k)) == 1 assert refine(exp(pi*I*2*(k + S.Half))) == -1 assert refine(exp(pi*I*2*(k + Rational(1, 4)))) == I assert refine(exp(pi*I*2*(k + Rational(3, 4)))) == -I assert exp(log(x)) == x assert exp(2*log(x)) == x**2 assert exp(pi*log(x)) == x**pi assert exp(17*log(x) + E*log(y)) == x**17 * y**E assert exp(x*log(x)) != x**x assert exp(sin(x)*log(x)) != x assert exp(3*log(x) + oo*x) == exp(oo*x) * x**3 assert exp(4*log(x)*log(y) + 3*log(x)) == x**3 * exp(4*log(x)*log(y)) assert exp(-oo, evaluate=False).is_finite is True assert exp(oo, evaluate=False).is_finite is False @_both_exp_pow def test_exp_period(): assert exp(I*pi*Rational(9, 4)) == exp(I*pi/4) assert exp(I*pi*Rational(46, 18)) == exp(I*pi*Rational(5, 9)) assert exp(I*pi*Rational(25, 7)) == exp(I*pi*Rational(-3, 7)) assert exp(I*pi*Rational(-19, 3)) == exp(-I*pi/3) assert exp(I*pi*Rational(37, 8)) - exp(I*pi*Rational(-11, 8)) == 0 assert exp(I*pi*Rational(-5, 3)) / exp(I*pi*Rational(11, 5)) * exp(I*pi*Rational(148, 15)) == 1 assert exp(2 - I*pi*Rational(17, 5)) == exp(2 + I*pi*Rational(3, 5)) assert exp(log(3) + I*pi*Rational(29, 9)) == 3 * exp(I*pi*Rational(-7, 9)) n = Symbol('n', integer=True) e = Symbol('e', even=True) assert exp(e*I*pi) == 1 assert exp((e + 1)*I*pi) == -1 assert exp((1 + 4*n)*I*pi/2) == I assert exp((-1 + 4*n)*I*pi/2) == -I @_both_exp_pow def test_exp_log(): x = Symbol("x", real=True) assert log(exp(x)) == x assert exp(log(x)) == x if not global_parameters.exp_is_pow: assert log(x).inverse() == exp assert exp(x).inverse() == log y = Symbol("y", polar=True) assert log(exp_polar(z)) == z assert exp(log(y)) == y @_both_exp_pow def test_exp_expand(): e = exp(log(Rational(2))*(1 + x) - log(Rational(2))*x) assert e.expand() == 2 assert exp(x + y) != exp(x)*exp(y) assert exp(x + y).expand() == exp(x)*exp(y) @_both_exp_pow def test_exp__as_base_exp(): assert exp(x).as_base_exp() == (E, x) assert exp(2*x).as_base_exp() == (E, 2*x) assert exp(x*y).as_base_exp() == (E, x*y) assert exp(-x).as_base_exp() == (E, -x) # Pow( *expr.as_base_exp() ) == expr invariant should hold assert E**x == exp(x) assert E**(2*x) == exp(2*x) assert E**(x*y) == exp(x*y) assert exp(x).base is S.Exp1 assert exp(x).exp == x @_both_exp_pow def test_exp_infinity(): assert exp(I*y) != nan assert refine(exp(I*oo)) is nan assert refine(exp(-I*oo)) is nan assert exp(y*I*oo) != nan assert exp(zoo) is nan x = Symbol('x', extended_real=True, finite=False) assert exp(x).is_complex is None @_both_exp_pow def test_exp_subs(): x = Symbol('x') e = (exp(3*log(x), evaluate=False)) # evaluates to x**3 assert e.subs(x**3, y**3) == e assert e.subs(x**2, 5) == e assert (x**3).subs(x**2, y) != y**Rational(3, 2) assert exp(exp(x) + exp(x**2)).subs(exp(exp(x)), y) == y * exp(exp(x**2)) assert exp(x).subs(E, y) == y**x x = symbols('x', real=True) assert exp(5*x).subs(exp(7*x), y) == y**Rational(5, 7) assert exp(2*x + 7).subs(exp(3*x), y) == y**Rational(2, 3) * exp(7) x = symbols('x', positive=True) assert exp(3*log(x)).subs(x**2, y) == y**Rational(3, 2) # differentiate between E and exp assert exp(exp(x + E)).subs(exp, 3) == 3**(3**(x + E)) assert exp(exp(x + E)).subs(exp, sin) == sin(sin(x + E)) assert exp(exp(x + E)).subs(E, 3) == 3**(3**(x + 3)) assert exp(3).subs(E, sin) == sin(3) def test_exp_adjoint(): assert adjoint(exp(x)) == exp(adjoint(x)) def test_exp_conjugate(): assert conjugate(exp(x)) == exp(conjugate(x)) @_both_exp_pow def test_exp_transpose(): assert transpose(exp(x)) == exp(transpose(x)) @_both_exp_pow def test_exp_rewrite(): assert exp(x).rewrite(sin) == sinh(x) + cosh(x) assert exp(x*I).rewrite(cos) == cos(x) + I*sin(x) assert exp(1).rewrite(cos) == sinh(1) + cosh(1) assert exp(1).rewrite(sin) == sinh(1) + cosh(1) assert exp(1).rewrite(sin) == sinh(1) + cosh(1) assert exp(x).rewrite(tanh) == (1 + tanh(x/2))/(1 - tanh(x/2)) assert exp(pi*I/4).rewrite(sqrt) == sqrt(2)/2 + sqrt(2)*I/2 assert exp(pi*I/3).rewrite(sqrt) == S.Half + sqrt(3)*I/2 if not global_parameters.exp_is_pow: assert exp(x*log(y)).rewrite(Pow) == y**x assert exp(log(x)*log(y)).rewrite(Pow) in [x**log(y), y**log(x)] assert exp(log(log(x))*y).rewrite(Pow) == log(x)**y n = Symbol('n', integer=True) assert Sum((exp(pi*I/2)/2)**n, (n, 0, oo)).rewrite(sqrt).doit() == Rational(4, 5) + I*2/5 assert Sum((exp(pi*I/4)/2)**n, (n, 0, oo)).rewrite(sqrt).doit() == 1/(1 - sqrt(2)*(1 + I)/4) assert (Sum((exp(pi*I/3)/2)**n, (n, 0, oo)).rewrite(sqrt).doit().cancel() == 4*I/(sqrt(3) + 3*I)) @_both_exp_pow def test_exp_leading_term(): assert exp(x).as_leading_term(x) == 1 assert exp(2 + x).as_leading_term(x) == exp(2) assert exp((2*x + 3) / (x+1)).as_leading_term(x) == exp(3) # The following tests are commented, since now SymPy returns the # original function when the leading term in the series expansion does # not exist. # raises(NotImplementedError, lambda: exp(1/x).as_leading_term(x)) # raises(NotImplementedError, lambda: exp((x + 1) / x**2).as_leading_term(x)) # raises(NotImplementedError, lambda: exp(x + 1/x).as_leading_term(x)) @_both_exp_pow def test_exp_taylor_term(): x = symbols('x') assert exp(x).taylor_term(1, x) == x assert exp(x).taylor_term(3, x) == x**3/6 assert exp(x).taylor_term(4, x) == x**4/24 assert exp(x).taylor_term(-1, x) is S.Zero def test_exp_MatrixSymbol(): A = MatrixSymbol("A", 2, 2) assert exp(A).has(exp) def test_exp_fdiff(): x = Symbol('x') raises(ArgumentIndexError, lambda: exp(x).fdiff(2)) def test_log_values(): assert log(nan) is nan assert log(oo) is oo assert log(-oo) is oo assert log(zoo) is zoo assert log(-zoo) is zoo assert log(0) is zoo assert log(1) == 0 assert log(-1) == I*pi assert log(E) == 1 assert log(-E).expand() == 1 + I*pi assert unchanged(log, pi) assert log(-pi).expand() == log(pi) + I*pi assert unchanged(log, 17) assert log(-17) == log(17) + I*pi assert log(I) == I*pi/2 assert log(-I) == -I*pi/2 assert log(17*I) == I*pi/2 + log(17) assert log(-17*I).expand() == -I*pi/2 + log(17) assert log(oo*I) is oo assert log(-oo*I) is oo assert log(0, 2) is zoo assert log(0, 5) is zoo assert exp(-log(3))**(-1) == 3 assert log(S.Half) == -log(2) assert log(2*3).func is log assert log(2*3**2).func is log def test_match_real_imag(): x, y = symbols('x,y', real=True) i = Symbol('i', imaginary=True) assert match_real_imag(S.One) == (1, 0) assert match_real_imag(I) == (0, 1) assert match_real_imag(3 - 5*I) == (3, -5) assert match_real_imag(-sqrt(3) + S.Half*I) == (-sqrt(3), S.Half) assert match_real_imag(x + y*I) == (x, y) assert match_real_imag(x*I + y*I) == (0, x + y) assert match_real_imag((x + y)*I) == (0, x + y) assert match_real_imag(Rational(-2, 3)*i*I) == (None, None) assert match_real_imag(1 - 2*i) == (None, None) assert match_real_imag(sqrt(2)*(3 - 5*I)) == (None, None) def test_log_exact(): # check for pi/2, pi/3, pi/4, pi/6, pi/8, pi/12; pi/5, pi/10: for n in range(-23, 24): if gcd(n, 24) != 1: assert log(exp(n*I*pi/24).rewrite(sqrt)) == n*I*pi/24 for n in range(-9, 10): assert log(exp(n*I*pi/10).rewrite(sqrt)) == n*I*pi/10 assert log(S.Half - I*sqrt(3)/2) == -I*pi/3 assert log(Rational(-1, 2) + I*sqrt(3)/2) == I*pi*Rational(2, 3) assert log(-sqrt(2)/2 - I*sqrt(2)/2) == -I*pi*Rational(3, 4) assert log(-sqrt(3)/2 - I*S.Half) == -I*pi*Rational(5, 6) assert log(Rational(-1, 4) + sqrt(5)/4 - I*sqrt(sqrt(5)/8 + Rational(5, 8))) == -I*pi*Rational(2, 5) assert log(sqrt(Rational(5, 8) - sqrt(5)/8) + I*(Rational(1, 4) + sqrt(5)/4)) == I*pi*Rational(3, 10) assert log(-sqrt(sqrt(2)/4 + S.Half) + I*sqrt(S.Half - sqrt(2)/4)) == I*pi*Rational(7, 8) assert log(-sqrt(6)/4 - sqrt(2)/4 + I*(-sqrt(6)/4 + sqrt(2)/4)) == -I*pi*Rational(11, 12) assert log(-1 + I*sqrt(3)) == log(2) + I*pi*Rational(2, 3) assert log(5 + 5*I) == log(5*sqrt(2)) + I*pi/4 assert log(sqrt(-12)) == log(2*sqrt(3)) + I*pi/2 assert log(-sqrt(6) + sqrt(2) - I*sqrt(6) - I*sqrt(2)) == log(4) - I*pi*Rational(7, 12) assert log(-sqrt(6-3*sqrt(2)) - I*sqrt(6+3*sqrt(2))) == log(2*sqrt(3)) - I*pi*Rational(5, 8) assert log(1 + I*sqrt(2-sqrt(2))/sqrt(2+sqrt(2))) == log(2/sqrt(sqrt(2) + 2)) + I*pi/8 assert log(cos(pi*Rational(7, 12)) + I*sin(pi*Rational(7, 12))) == I*pi*Rational(7, 12) assert log(cos(pi*Rational(6, 5)) + I*sin(pi*Rational(6, 5))) == I*pi*Rational(-4, 5) assert log(5*(1 + I)/sqrt(2)) == log(5) + I*pi/4 assert log(sqrt(2)*(-sqrt(3) + 1 - sqrt(3)*I - I)) == log(4) - I*pi*Rational(7, 12) assert log(-sqrt(2)*(1 - I*sqrt(3))) == log(2*sqrt(2)) + I*pi*Rational(2, 3) assert log(sqrt(3)*I*(-sqrt(6 - 3*sqrt(2)) - I*sqrt(3*sqrt(2) + 6))) == log(6) - I*pi/8 zero = (1 + sqrt(2))**2 - 3 - 2*sqrt(2) assert log(zero - I*sqrt(3)) == log(sqrt(3)) - I*pi/2 assert unchanged(log, zero + I*zero) or log(zero + zero*I) is zoo # bail quickly if no obvious simplification is possible: assert unchanged(log, (sqrt(2)-1/sqrt(sqrt(3)+I))**1000) # beware of non-real coefficients assert unchanged(log, sqrt(2-sqrt(5))*(1 + I)) def test_log_base(): assert log(1, 2) == 0 assert log(2, 2) == 1 assert log(3, 2) == log(3)/log(2) assert log(6, 2) == 1 + log(3)/log(2) assert log(6, 3) == 1 + log(2)/log(3) assert log(2**3, 2) == 3 assert log(3**3, 3) == 3 assert log(5, 1) is zoo assert log(1, 1) is nan assert log(Rational(2, 3), 10) == log(Rational(2, 3))/log(10) assert log(Rational(2, 3), Rational(1, 3)) == -log(2)/log(3) + 1 assert log(Rational(2, 3), Rational(2, 5)) == \ log(Rational(2, 3))/log(Rational(2, 5)) # issue 17148 assert log(Rational(8, 3), 2) == -log(3)/log(2) + 3 def test_log_symbolic(): assert log(x, exp(1)) == log(x) assert log(exp(x)) != x assert log(x, exp(1)) == log(x) assert log(x*y) != log(x) + log(y) assert log(x/y).expand() != log(x) - log(y) assert log(x/y).expand(force=True) == log(x) - log(y) assert log(x**y).expand() != y*log(x) assert log(x**y).expand(force=True) == y*log(x) assert log(x, 2) == log(x)/log(2) assert log(E, 2) == 1/log(2) p, q = symbols('p,q', positive=True) r = Symbol('r', real=True) assert log(p**2) != 2*log(p) assert log(p**2).expand() == 2*log(p) assert log(x**2).expand() != 2*log(x) assert log(p**q) != q*log(p) assert log(exp(p)) == p assert log(p*q) != log(p) + log(q) assert log(p*q).expand() == log(p) + log(q) assert log(-sqrt(3)) == log(sqrt(3)) + I*pi assert log(-exp(p)) != p + I*pi assert log(-exp(x)).expand() != x + I*pi assert log(-exp(r)).expand() == r + I*pi assert log(x**y) != y*log(x) assert (log(x**-5)**-1).expand() != -1/log(x)/5 assert (log(p**-5)**-1).expand() == -1/log(p)/5 assert log(-x).func is log and log(-x).args[0] == -x assert log(-p).func is log and log(-p).args[0] == -p def test_log_exp(): assert log(exp(4*I*pi)) == 0 # exp evaluates assert log(exp(-5*I*pi)) == I*pi # exp evaluates assert log(exp(I*pi*Rational(19, 4))) == I*pi*Rational(3, 4) assert log(exp(I*pi*Rational(25, 7))) == I*pi*Rational(-3, 7) assert log(exp(-5*I)) == -5*I + 2*I*pi @_both_exp_pow def test_exp_assumptions(): r = Symbol('r', real=True) i = Symbol('i', imaginary=True) for e in exp, exp_polar: assert e(x).is_real is None assert e(x).is_imaginary is None assert e(i).is_real is None assert e(i).is_imaginary is None assert e(r).is_real is True assert e(r).is_imaginary is False assert e(re(x)).is_extended_real is True assert e(re(x)).is_imaginary is False assert Pow(E, I*pi, evaluate=False).is_imaginary == False assert Pow(E, 2*I*pi, evaluate=False).is_imaginary == False assert Pow(E, I*pi/2, evaluate=False).is_imaginary == True assert Pow(E, I*pi/3, evaluate=False).is_imaginary is None assert exp(0, evaluate=False).is_algebraic a = Symbol('a', algebraic=True) an = Symbol('an', algebraic=True, nonzero=True) r = Symbol('r', rational=True) rn = Symbol('rn', rational=True, nonzero=True) assert exp(a).is_algebraic is None assert exp(an).is_algebraic is False assert exp(pi*r).is_algebraic is None assert exp(pi*rn).is_algebraic is False assert exp(0, evaluate=False).is_algebraic is True assert exp(I*pi/3, evaluate=False).is_algebraic is True assert exp(I*pi*r, evaluate=False).is_algebraic is True @_both_exp_pow def test_exp_AccumBounds(): assert exp(AccumBounds(1, 2)) == AccumBounds(E, E**2) def test_log_assumptions(): p = symbols('p', positive=True) n = symbols('n', negative=True) z = symbols('z', zero=True) x = symbols('x', infinite=True, extended_positive=True) assert log(z).is_positive is False assert log(x).is_extended_positive is True assert log(2) > 0 assert log(1, evaluate=False).is_zero assert log(1 + z).is_zero assert log(p).is_zero is None assert log(n).is_zero is False assert log(0.5).is_negative is True assert log(exp(p) + 1).is_positive assert log(1, evaluate=False).is_algebraic assert log(42, evaluate=False).is_algebraic is False assert log(1 + z).is_rational def test_log_hashing(): assert x != log(log(x)) assert hash(x) != hash(log(log(x))) assert log(x) != log(log(log(x))) e = 1/log(log(x) + log(log(x))) assert e.base.func is log e = 1/log(log(x) + log(log(log(x)))) assert e.base.func is log e = log(log(x)) assert e.func is log assert x.func is not log assert hash(log(log(x))) != hash(x) assert e != x def test_log_sign(): assert sign(log(2)) == 1 def test_log_expand_complex(): assert log(1 + I).expand(complex=True) == log(2)/2 + I*pi/4 assert log(1 - sqrt(2)).expand(complex=True) == log(sqrt(2) - 1) + I*pi def test_log_apply_evalf(): value = (log(3)/log(2) - 1).evalf() assert value.epsilon_eq(Float("0.58496250072115618145373")) def test_log_leading_term(): p = Symbol('p') # Test for STEP 3 assert log(1 + x + x**2).as_leading_term(x, cdir=1) == x # Test for STEP 4 assert log(2*x).as_leading_term(x, cdir=1) == log(x) + log(2) assert log(2*x).as_leading_term(x, cdir=-1) == log(x) + log(2) assert log(-2*x).as_leading_term(x, cdir=1, logx=p) == p + log(2) + I*pi assert log(-2*x).as_leading_term(x, cdir=-1, logx=p) == p + log(2) - I*pi # Test for STEP 5 assert log(-2*x + (3 - I)*x**2).as_leading_term(x, cdir=1) == log(x) + log(2) - I*pi assert log(-2*x + (3 - I)*x**2).as_leading_term(x, cdir=-1) == log(x) + log(2) - I*pi assert log(2*x + (3 - I)*x**2).as_leading_term(x, cdir=1) == log(x) + log(2) assert log(2*x + (3 - I)*x**2).as_leading_term(x, cdir=-1) == log(x) + log(2) - 2*I*pi assert log(-1 + x - I*x**2 + I*x**3).as_leading_term(x, cdir=1) == -I*pi assert log(-1 + x - I*x**2 + I*x**3).as_leading_term(x, cdir=-1) == -I*pi assert log(-1/(1 - x)).as_leading_term(x, cdir=1) == I*pi assert log(-1/(1 - x)).as_leading_term(x, cdir=-1) == I*pi def test_log_nseries(): p = Symbol('p') assert log(1/x)._eval_nseries(x, 4, logx=-p, cdir=1) == p assert log(1/x)._eval_nseries(x, 4, logx=-p, cdir=-1) == p + 2*I*pi assert log(x - 1)._eval_nseries(x, 4, None, I) == I*pi - x - x**2/2 - x**3/3 + O(x**4) assert log(x - 1)._eval_nseries(x, 4, None, -I) == -I*pi - x - x**2/2 - x**3/3 + O(x**4) assert log(I*x + I*x**3 - 1)._eval_nseries(x, 3, None, 1) == I*pi - I*x + x**2/2 + O(x**3) assert log(I*x + I*x**3 - 1)._eval_nseries(x, 3, None, -1) == -I*pi - I*x + x**2/2 + O(x**3) assert log(I*x**2 + I*x**3 - 1)._eval_nseries(x, 3, None, 1) == I*pi - I*x**2 + O(x**3) assert log(I*x**2 + I*x**3 - 1)._eval_nseries(x, 3, None, -1) == I*pi - I*x**2 + O(x**3) assert log(2*x + (3 - I)*x**2)._eval_nseries(x, 3, None, 1) == log(2) + log(x) + \ x*(S(3)/2 - I/2) + x**2*(-1 + 3*I/4) + O(x**3) assert log(2*x + (3 - I)*x**2)._eval_nseries(x, 3, None, -1) == -2*I*pi + log(2) + \ log(x) - x*(-S(3)/2 + I/2) + x**2*(-1 + 3*I/4) + O(x**3) assert log(-2*x + (3 - I)*x**2)._eval_nseries(x, 3, None, 1) == -I*pi + log(2) + log(x) + \ x*(-S(3)/2 + I/2) + x**2*(-1 + 3*I/4) + O(x**3) assert log(-2*x + (3 - I)*x**2)._eval_nseries(x, 3, None, -1) == -I*pi + log(2) + log(x) - \ x*(S(3)/2 - I/2) + x**2*(-1 + 3*I/4) + O(x**3) assert log(sqrt(-I*x**2 - 3)*sqrt(-I*x**2 - 1) - 2)._eval_nseries(x, 3, None, 1) == -I*pi + \ log(sqrt(3) + 2) + I*x**2*(-2 + 4*sqrt(3)/3) + O(x**3) assert log(-1/(1 - x))._eval_nseries(x, 3, None, 1) == I*pi + x + x**2/2 + O(x**3) assert log(-1/(1 - x))._eval_nseries(x, 3, None, -1) == I*pi + x + x**2/2 + O(x**3) def test_log_series(): # Note Series at infinities other than oo/-oo were introduced as a part of # pull request 23798. Refer https://github.com/sympy/sympy/pull/23798 for # more information. expr1 = log(1 + x) expr2 = log(x + sqrt(x**2 + 1)) assert expr1.series(x, x0=I*oo, n=4) == 1/(3*x**3) - 1/(2*x**2) + 1/x + \ I*pi/2 - log(I/x) + O(x**(-4), (x, oo*I)) assert expr1.series(x, x0=-I*oo, n=4) == 1/(3*x**3) - 1/(2*x**2) + 1/x - \ I*pi/2 - log(-I/x) + O(x**(-4), (x, -oo*I)) assert expr2.series(x, x0=I*oo, n=4) == 1/(4*x**2) + I*pi/2 + log(2) - \ log(I/x) + O(x**(-4), (x, oo*I)) assert expr2.series(x, x0=-I*oo, n=4) == -1/(4*x**2) - I*pi/2 - log(2) + \ log(-I/x) + O(x**(-4), (x, -oo*I)) def test_log_expand(): w = Symbol("w", positive=True) e = log(w**(log(5)/log(3))) assert e.expand() == log(5)/log(3) * log(w) x, y, z = symbols('x,y,z', positive=True) assert log(x*(y + z)).expand(mul=False) == log(x) + log(y + z) assert log(log(x**2)*log(y*z)).expand() in [log(2*log(x)*log(y) + 2*log(x)*log(z)), log(log(x)*log(z) + log(y)*log(x)) + log(2), log((log(y) + log(z))*log(x)) + log(2)] assert log(x**log(x**2)).expand(deep=False) == log(x)*log(x**2) assert log(x**log(x**2)).expand() == 2*log(x)**2 x, y = symbols('x,y') assert log(x*y).expand(force=True) == log(x) + log(y) assert log(x**y).expand(force=True) == y*log(x) assert log(exp(x)).expand(force=True) == x # there's generally no need to expand out logs since this requires # factoring and if simplification is sought, it's cheaper to put # logs together than it is to take them apart. assert log(2*3**2).expand() != 2*log(3) + log(2) @XFAIL def test_log_expand_fail(): x, y, z = symbols('x,y,z', positive=True) assert (log(x*(y + z))*(x + y)).expand(mul=True, log=True) == y*log( x) + y*log(y + z) + z*log(x) + z*log(y + z) def test_log_simplify(): x = Symbol("x", positive=True) assert log(x**2).expand() == 2*log(x) assert expand_log(log(x**(2 + log(2)))) == (2 + log(2))*log(x) z = Symbol('z') assert log(sqrt(z)).expand() == log(z)/2 assert expand_log(log(z**(log(2) - 1))) == (log(2) - 1)*log(z) assert log(z**(-1)).expand() != -log(z) assert log(z**(x/(x+1))).expand() == x*log(z)/(x + 1) def test_log_AccumBounds(): assert log(AccumBounds(1, E)) == AccumBounds(0, 1) assert log(AccumBounds(0, E)) == AccumBounds(-oo, 1) assert log(AccumBounds(-1, E)) == S.NaN assert log(AccumBounds(0, oo)) == AccumBounds(-oo, oo) assert log(AccumBounds(-oo, 0)) == S.NaN assert log(AccumBounds(-oo, oo)) == S.NaN @_both_exp_pow def test_lambertw(): k = Symbol('k') assert LambertW(x, 0) == LambertW(x) assert LambertW(x, 0, evaluate=False) != LambertW(x) assert LambertW(0) == 0 assert LambertW(E) == 1 assert LambertW(-1/E) == -1 assert LambertW(-log(2)/2) == -log(2) assert LambertW(oo) is oo assert LambertW(0, 1) is -oo assert LambertW(0, 42) is -oo assert LambertW(-pi/2, -1) == -I*pi/2 assert LambertW(-1/E, -1) == -1 assert LambertW(-2*exp(-2), -1) == -2 assert LambertW(2*log(2)) == log(2) assert LambertW(-pi/2) == I*pi/2 assert LambertW(exp(1 + E)) == E assert LambertW(x**2).diff(x) == 2*LambertW(x**2)/x/(1 + LambertW(x**2)) assert LambertW(x, k).diff(x) == LambertW(x, k)/x/(1 + LambertW(x, k)) assert LambertW(sqrt(2)).evalf(30).epsilon_eq( Float("0.701338383413663009202120278965", 30), 1e-29) assert re(LambertW(2, -1)).evalf().epsilon_eq(Float("-0.834310366631110")) assert LambertW(-1).is_real is False # issue 5215 assert LambertW(2, evaluate=False).is_real p = Symbol('p', positive=True) assert LambertW(p, evaluate=False).is_real assert LambertW(p - 1, evaluate=False).is_real is None assert LambertW(-p - 2/S.Exp1, evaluate=False).is_real is False assert LambertW(S.Half, -1, evaluate=False).is_real is False assert LambertW(Rational(-1, 10), -1, evaluate=False).is_real assert LambertW(-10, -1, evaluate=False).is_real is False assert LambertW(-2, 2, evaluate=False).is_real is False assert LambertW(0, evaluate=False).is_algebraic na = Symbol('na', nonzero=True, algebraic=True) assert LambertW(na).is_algebraic is False assert LambertW(p).is_zero is False n = Symbol('n', negative=True) assert LambertW(n).is_zero is False def test_issue_5673(): e = LambertW(-1) assert e.is_comparable is False assert e.is_positive is not True e2 = 1 - 1/(1 - exp(-1000)) assert e2.is_positive is not True e3 = -2 + exp(exp(LambertW(log(2)))*LambertW(log(2))) assert e3.is_nonzero is not True def test_log_fdiff(): x = Symbol('x') raises(ArgumentIndexError, lambda: log(x).fdiff(2)) def test_log_taylor_term(): x = symbols('x') assert log(x).taylor_term(0, x) == x assert log(x).taylor_term(1, x) == -x**2/2 assert log(x).taylor_term(4, x) == x**5/5 assert log(x).taylor_term(-1, x) is S.Zero def test_exp_expand_NC(): A, B, C = symbols('A,B,C', commutative=False) assert exp(A + B).expand() == exp(A + B) assert exp(A + B + C).expand() == exp(A + B + C) assert exp(x + y).expand() == exp(x)*exp(y) assert exp(x + y + z).expand() == exp(x)*exp(y)*exp(z) @_both_exp_pow def test_as_numer_denom(): n = symbols('n', negative=True) assert exp(x).as_numer_denom() == (exp(x), 1) assert exp(-x).as_numer_denom() == (1, exp(x)) assert exp(-2*x).as_numer_denom() == (1, exp(2*x)) assert exp(-2).as_numer_denom() == (1, exp(2)) assert exp(n).as_numer_denom() == (1, exp(-n)) assert exp(-n).as_numer_denom() == (exp(-n), 1) assert exp(-I*x).as_numer_denom() == (1, exp(I*x)) assert exp(-I*n).as_numer_denom() == (1, exp(I*n)) assert exp(-n).as_numer_denom() == (exp(-n), 1) @_both_exp_pow def test_polar(): x, y = symbols('x y', polar=True) assert abs(exp_polar(I*4)) == 1 assert abs(exp_polar(0)) == 1 assert abs(exp_polar(2 + 3*I)) == exp(2) assert exp_polar(I*10).n() == exp_polar(I*10) assert log(exp_polar(z)) == z assert log(x*y).expand() == log(x) + log(y) assert log(x**z).expand() == z*log(x) assert exp_polar(3).exp == 3 # Compare exp(1.0*pi*I). assert (exp_polar(1.0*pi*I).n(n=5)).as_real_imag()[1] >= 0 assert exp_polar(0).is_rational is True # issue 8008 def test_exp_summation(): w = symbols("w") m, n, i, j = symbols("m n i j") expr = exp(Sum(w*i, (i, 0, n), (j, 0, m))) assert expr.expand() == Product(exp(w*i), (i, 0, n), (j, 0, m)) def test_log_product(): from sympy.abc import n, m i, j = symbols('i,j', positive=True, integer=True) x, y = symbols('x,y', positive=True) z = symbols('z', real=True) w = symbols('w') expr = log(Product(x**i, (i, 1, n))) assert simplify(expr) == expr assert expr.expand() == Sum(i*log(x), (i, 1, n)) expr = log(Product(x**i*y**j, (i, 1, n), (j, 1, m))) assert simplify(expr) == expr assert expr.expand() == Sum(i*log(x) + j*log(y), (i, 1, n), (j, 1, m)) expr = log(Product(-2, (n, 0, 4))) assert simplify(expr) == expr assert expr.expand() == expr assert expr.expand(force=True) == Sum(log(-2), (n, 0, 4)) expr = log(Product(exp(z*i), (i, 0, n))) assert expr.expand() == Sum(z*i, (i, 0, n)) expr = log(Product(exp(w*i), (i, 0, n))) assert expr.expand() == expr assert expr.expand(force=True) == Sum(w*i, (i, 0, n)) expr = log(Product(i**2*abs(j), (i, 1, n), (j, 1, m))) assert expr.expand() == Sum(2*log(i) + log(j), (i, 1, n), (j, 1, m)) @XFAIL def test_log_product_simplify_to_sum(): from sympy.abc import n, m i, j = symbols('i,j', positive=True, integer=True) x, y = symbols('x,y', positive=True) assert simplify(log(Product(x**i, (i, 1, n)))) == Sum(i*log(x), (i, 1, n)) assert simplify(log(Product(x**i*y**j, (i, 1, n), (j, 1, m)))) == \ Sum(i*log(x) + j*log(y), (i, 1, n), (j, 1, m)) def test_issue_8866(): assert simplify(log(x, 10, evaluate=False)) == simplify(log(x, 10)) assert expand_log(log(x, 10, evaluate=False)) == expand_log(log(x, 10)) y = Symbol('y', positive=True) l1 = log(exp(y), exp(10)) b1 = log(exp(y), exp(5)) l2 = log(exp(y), exp(10), evaluate=False) b2 = log(exp(y), exp(5), evaluate=False) assert simplify(log(l1, b1)) == simplify(log(l2, b2)) assert expand_log(log(l1, b1)) == expand_log(log(l2, b2)) def test_log_expand_factor(): assert (log(18)/log(3) - 2).expand(factor=True) == log(2)/log(3) assert (log(12)/log(2)).expand(factor=True) == log(3)/log(2) + 2 assert (log(15)/log(3)).expand(factor=True) == 1 + log(5)/log(3) assert (log(2)/(-log(12) + log(24))).expand(factor=True) == 1 assert expand_log(log(12), factor=True) == log(3) + 2*log(2) assert expand_log(log(21)/log(7), factor=False) == log(3)/log(7) + 1 assert expand_log(log(45)/log(5) + log(20), factor=False) == \ 1 + 2*log(3)/log(5) + log(20) assert expand_log(log(45)/log(5) + log(26), factor=True) == \ log(2) + log(13) + (log(5) + 2*log(3))/log(5) def test_issue_9116(): n = Symbol('n', positive=True, integer=True) assert log(n).is_nonnegative is True def test_issue_18473(): assert exp(x*log(cos(1/x))).as_leading_term(x) == S.NaN assert exp(x*log(tan(1/x))).as_leading_term(x) == S.NaN assert log(cos(1/x)).as_leading_term(x) == S.NaN assert log(tan(1/x)).as_leading_term(x) == S.NaN assert log(cos(1/x) + 2).as_leading_term(x) == AccumBounds(0, log(3)) assert exp(x*log(cos(1/x) + 2)).as_leading_term(x) == 1 assert log(cos(1/x) - 2).as_leading_term(x) == S.NaN assert exp(x*log(cos(1/x) - 2)).as_leading_term(x) == S.NaN assert log(cos(1/x) + 1).as_leading_term(x) == AccumBounds(-oo, log(2)) assert exp(x*log(cos(1/x) + 1)).as_leading_term(x) == AccumBounds(0, 1) assert log(sin(1/x)**2).as_leading_term(x) == AccumBounds(-oo, 0) assert exp(x*log(sin(1/x)**2)).as_leading_term(x) == AccumBounds(0, 1) assert log(tan(1/x)**2).as_leading_term(x) == AccumBounds(-oo, oo) assert exp(2*x*(log(tan(1/x)**2))).as_leading_term(x) == AccumBounds(0, oo)